statement stringlengths 1 3.67k | proof stringlengths 0 21.4k | type stringclasses 9
values | symbolic_name stringlengths 1 41 | library stringclasses 25
values | filename stringclasses 171
values | imports listlengths 0 0 | deps listlengths 0 64 | docstring stringclasses 1
value | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
Type3 := Type. | definition | Type3 | matita/components/ng_kernel | matita/components/ng_kernel/bug_universi.ma | [] | [] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | ||
Type2 : Type3 := Type. | definition | Type2 | matita/components/ng_kernel | matita/components/ng_kernel/bug_universi.ma | [] | [
"Type3"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | ||
Type1 : Type2 := Type. | definition | Type1 | matita/components/ng_kernel | matita/components/ng_kernel/bug_universi.ma | [] | [
"Type2"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | ||
t: Type2 | ≝
k: Type3 -> t. | inductive | t | matita/components/ng_kernel | matita/components/ng_kernel/bug_universi.ma | [] | [
"Type2",
"Type3"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
f n :=
match n with
[ O => O
| S m => g m
]
and g m :=
match m with
[ O => O
| S n => f n
]. | let rec | f | matita/components/ng_kernel | matita/components/ng_kernel/test.ma | [] | [] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | ||
hole : ∀A:Type.A → A | ≝ λA.λx.x. | definition | hole | matita/components/ng_refiner | matita/components/ng_refiner/esempio.ma | [] | [] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
id : ∀A:Type.A → A | ≝ λA.λx.x.
(* Common case in dama, reduction with metas
inductive list : Type := nil : list | cons : nat -> list -> list.
let rec len l := match l with [ nil => O | cons _ l => S (len l) ].
axiom lt : nat -> nat -> Prop.
axiom foo : ∀x. Not (lt (hole ? x) (hole ? O)) = (lt x (len nil) -> False).
*)
(* meta1 Vs meta2... | definition | id | matita/components/ng_refiner | matita/components/ng_refiner/esempio.ma | [] | [] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
foo: (100+111) = (100+110).
(id ?(id ?(id ?(id ? (100+100))))) =
(id ?(id ?(id ?(id ? (99+100))))).[3:
apply (refl_eq nat (id ?(id ?(id ?(id ? (98+102+?)))))); | axiom | foo | matita/components/ng_refiner | matita/components/ng_refiner/esempio.ma | [] | [
"id",
"nat"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | ||
foo: (λx,y.(λz. z (x+y) + z x) (λw:nat.hole ? w)) = λx,y.x. | axiom | foo | matita/components/ng_refiner | matita/components/ng_refiner/esempio.ma | [] | [
"hole",
"nat"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | ||
foo: (λx,y.(λz. z x + z (x+y)) (λw:nat.hole ? w)) = λx,y.x. | axiom | foo | matita/components/ng_refiner | matita/components/ng_refiner/esempio.ma | [] | [
"hole",
"nat"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | ||
foo: (λx,y.(λz. z x + z y) (λw:nat.hole ? w)) = λx,y.x. | axiom | foo | matita/components/ng_refiner | matita/components/ng_refiner/esempio.ma | [] | [
"hole",
"nat"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | ||
nat : Type[0] | ≝ O: nat | S: nat → nat. | inductive | nat | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
test1: Type[1]. | axiom | test1 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | ||
test2: Type[1] → Type[1]. | axiom | test2 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | ||
test3: Prop → Type[1] → CProp[1] → Type[1] → Type[2]. | axiom | test3 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | ||
test4: ∀A:Type[1]. A → ∀B:Type[1]. B → ∀C:Prop. C → Type[1]. | axiom | test4 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | ||
test4': ∀C:Prop. C → C. | axiom | test4' | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | ||
test4'': ∀C:Prop. C → nat. | axiom | test4'' | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [
"nat"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | ||
test4''': ∀C:Type[1]. C. | axiom | test4''' | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | ||
test4_5: (∀A:Type[0].A) → nat. | axiom | test4_5 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [
"nat"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | ||
test5: (Type[1] → Type[1]) → Type[1]. | axiom | test5 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | ||
test6: Type[1] → Prop. | axiom | test6 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | ||
dtest1: Type[1] | ≝ nat → nat. | definition | dtest1 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [
"nat"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
dtest2: Type[2] | ≝ ∀A:Type[1]. A → A. | definition | dtest2 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
dtest3: Type[1] → Type[1] | ≝ λA:Type[1]. A → A. | definition | dtest3 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
dtest4: Type[1] → Type[1] | ≝ λA:Type[1].dtest3 A. | definition | dtest4 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [
"dtest3"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
dtest5: Type[1] → Type[1] | ≝ dtest3. | definition | dtest5 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [
"dtest3"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
dtest6: Type[1] | ≝ dtest3 nat. | definition | dtest6 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [
"dtest3",
"nat"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
True : Prop | ≝ I: True. | inductive | True | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
dtest7: Prop | ≝ True → True. | definition | dtest7 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [
"True"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
dtest8: Type[1] | ≝ dtest3 True. | definition | dtest8 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [
"True",
"dtest3"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
dtest9: Type[1] | ≝ dtest3 Prop. | definition | dtest9 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [
"dtest3"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
dtest10: Type[1] → Type[1] → Type[1] | ≝ λX,Y.X. | definition | dtest10 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
dtest11: Type[1] → nat → Type[1] → Type[1] | ≝ λ_:Type[1].λ_:nat.λX:Type[1]. X → nat → test1. | definition | dtest11 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [
"nat",
"test1"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
dtest12 | ≝ λ_:Type[1].λ_:nat.λX:Type[1]. X → nat → test1. | definition | dtest12 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [
"nat",
"test1"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
dtest13 | ≝ dtest3 nat → dtest3 True → dtest3 Prop → nat. | definition | dtest13 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [
"True",
"dtest3",
"nat"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
dtest14 | ≝ λX:Type[2]. X → X. | definition | dtest14 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
dtest15 | ≝ ∀T:Type[1] → Type[1]. T nat → T nat. | definition | dtest15 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [
"nat"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
dtest16 | ≝ ∀T:Type[1]. T → nat. | definition | dtest16 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [
"nat"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
dtest17 | ≝ ∀T:dtest14 Type[1]. T nat → dtest14 nat → dtest14 nat. | definition | dtest17 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [
"dtest14",
"nat"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
dtest18 | ≝ λA,B:Type[0].λn:nat.λC:Type[0].A. | definition | dtest18 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [
"nat"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
dtest19 | ≝ dtest18 nat True O nat → dtest18 nat nat O nat. | definition | dtest19 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [
"True",
"dtest18",
"nat"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
dtest20 | ≝ test5 test2. | definition | dtest20 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [
"test2",
"test5"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
ttest1 | ≝ λx:nat.x. | definition | ttest1 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [
"nat"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
ttest2 | ≝ λT:Type[1].λx:T.x. | definition | ttest2 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
ttest3 | ≝ λT:Type[1].λx:T.let y ≝ x in y. | definition | ttest3 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
ttest4 | ≝ λT:Type[1].let y ≝ T in λx:y.x. | definition | ttest4 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
ttest6 | ≝ ttest4 nat. | definition | ttest6 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [
"nat",
"ttest4"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
ttest7 | ≝ λf:∀X:Type[1].X. f (nat → nat) O. | definition | ttest7 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [
"nat"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
ttest8 | ≝ λf:∀X:Type[1].X. f (True → True) I. | definition | ttest8 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [
"True"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
ttest9 | ≝ λf:∀X:Type[1].X. f (True → nat) I. | definition | ttest9 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [
"True",
"nat"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
ttest10 | ≝ λf:∀X:Type[1].X. f (True → nat → nat) I O. | definition | ttest10 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [
"True",
"nat"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
ttest11_aux | ≝ λS:Type[1]. S → nat. | definition | ttest11_aux | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [
"nat"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
ttest11 | ≝ λf:ttest11_aux True. f I. | definition | ttest11 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [
"True",
"ttest11_aux"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
ttest12 | ≝ λf:True → nat. f I. | definition | ttest12 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [
"True",
"nat"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
rtest1 (n:nat) : nat | ≝
match n with
[ O ⇒ O
| S m ⇒ rtest1 m ]. | let rec | rtest1 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [
"nat"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
f (n:nat) : nat | ≝
match n with
[ O ⇒ O
| S m ⇒ g m ]
and g (n:nat) : nat ≝
match n with
[ O ⇒ O
| S m ⇒ f m ].
(*BUG: pattern matching patterns when arguments have been deleted from
the constructor are wrong *)
coinductive stream: Type[0] ≝ scons : nat → stream → stream.
let corec div (n:nat) : stream ≝ scons n (div (S n)). | let rec | f | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [
"div",
"nat"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
plus: nat → nat → nat. | axiom | plus | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [
"nat"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | ||
rtest2 : nat → stream → nat | ≝
λm,s. match s with [ scons n l ⇒ plus m n ]. | definition | rtest2 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [
"nat",
"plus"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
meee: Type[0] → Type[0] | ≝ . | inductive | meee | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
T1 : (Type[0] → Type[0]) → ∀B:Type[0]. nat → Type[0] → Type[0] | ≝ . | inductive | T1 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [
"nat"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
T2 : (Type[0] → Type[0]) → ∀B:Type[0]. B → Type[0] → Type[0] | ≝ . | inductive | T2 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
T3 : (Type[0] → Type[0]) → CProp[2] | ≝ . | inductive | T3 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
erase | ≝ λX:Type[0].Type[0]. | definition | erase | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
lt: nat → nat → Prop. | axiom | lt | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [
"nat"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | ||
myvect (A: Type[0]) (b:nat) : nat → Type[0] | ≝
myemptyv : myvect A b O
| mycons: ∀n. lt n b → A → myvect A b n → myvect A b (S n). | inductive | myvect | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [
"lt",
"nat"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
False: Prop | ≝ . | inductive | False | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
Empty: Type[0] | ≝ . | inductive | Empty | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
bool: Type[0] | ≝ true: bool | false:bool. | inductive | bool | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
eq (A:Type[1]) (a:A) : A → Prop | ≝ refl: eq A a a. | inductive | eq | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
cast_bug1 | ≝
λH:eq Type[0] bool nat. S (match H return λA:Type[0].λ_.A with [ refl ⇒ true ]). | definition | cast_bug1 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [
"bool",
"eq",
"nat"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
cast_bug2 | ≝
match true return λb.match b with [ true ⇒ nat → nat | false ⇒ bool ] with
[ true ⇒ S | false ⇒ false ]
O. | definition | cast_bug2 | matita/matita/lib | matita/matita/lib/extraction.ma | [] | [
"bool",
"nat"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
hint_declaration_Type0 | ≝ λA:Type[0] .λa,b:A.Prop. | definition | hint_declaration_Type0 | matita/matita/lib | matita/matita/lib/hints_declaration.ma | [] | [] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
hint_declaration_Type1 | ≝ λA:Type[1].λa,b:A.Prop. | definition | hint_declaration_Type1 | matita/matita/lib | matita/matita/lib/hints_declaration.ma | [] | [] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
hint_declaration_Type2 | ≝ λa,b:Type[2].Prop. | definition | hint_declaration_Type2 | matita/matita/lib | matita/matita/lib/hints_declaration.ma | [] | [] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
hint_declaration_CProp0 | ≝ λA:CProp[0].λa,b:A.Prop. | definition | hint_declaration_CProp0 | matita/matita/lib | matita/matita/lib/hints_declaration.ma | [] | [] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
hint_declaration_CProp1 | ≝ λA:CProp[1].λa,b:A.Prop. | definition | hint_declaration_CProp1 | matita/matita/lib | matita/matita/lib/hints_declaration.ma | [] | [] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
hint_declaration_CProp2 | ≝ λa,b:CProp[2].Prop. | definition | hint_declaration_CProp2 | matita/matita/lib | matita/matita/lib/hints_declaration.ma | [] | [] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
boh: (False → False) = True. | axiom | boh | matita/matita/lib | matita/matita/lib/inconsistent.ma | [] | [
"False",
"True"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | ||
J: False → False | ≝ λf. match f in False with [ ]. | definition | J | matita/matita/lib | matita/matita/lib/inconsistent.ma | [] | [
"False"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
I2: True | ≝ match boh with [ refl ⇒ J ]. | definition | I2 | matita/matita/lib | matita/matita/lib/inconsistent.ma | [] | [
"True",
"boh"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
err (u: True): False | ≝ match u with [ I ⇒ err I2 ]. | let rec | err | matita/matita/lib | matita/matita/lib/inconsistent.ma | [] | [
"False",
"I2",
"True"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
oops: False ≝ err I. | lemma | oops | matita/matita/lib | matita/matita/lib/inconsistent.ma | [] | [
"False",
"err"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | ||
inconsistent: Type[U] | ≝ Type[U]. | definition | inconsistent | matita/matita/lib | matita/matita/lib/self_typing.ma | [] | [] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
sameF_upto: nat → ∀A.relation(nat→A) | ≝
λk.λA.λf,g.∀i. i < k → f i = g i. | definition | sameF_upto | matita/matita/lib/arithmetics | matita/matita/lib/arithmetics/bigops.ma | [] | [
"nat",
"relation"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
sameF_p: nat → (nat → bool) →∀A.relation(nat→A) | ≝
λk,p,A,f,g.∀i. i < k → p i = true → f i = g i. | definition | sameF_p | matita/matita/lib/arithmetics | matita/matita/lib/arithmetics/bigops.ma | [] | [
"bool",
"nat",
"relation"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
sameF_upto_le: ∀A,f,g,n,m.
n ≤m → sameF_upto m A f g → sameF_upto n A f g. | #A #f #g #n #m #lenm #samef #i #ltin @samef /2 by lt_to_le_to_lt/
qed. | lemma | sameF_upto_le | matita/matita/lib/arithmetics | matita/matita/lib/arithmetics/bigops.ma | [] | [
"lt_to_le_to_lt",
"sameF_upto"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
sameF_p_le: ∀A,p,f,g,n,m.
n ≤m → sameF_p m p A f g → sameF_p n p A f g. | #A #p #f #g #n #m #lenm #samef #i #ltin #pi @samef /2 by lt_to_le_to_lt/
qed. | lemma | sameF_p_le | matita/matita/lib/arithmetics | matita/matita/lib/arithmetics/bigops.ma | [] | [
"lt_to_le_to_lt",
"sameF_p"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
prodF | ≝
λA,B.λf:nat→A.λg:nat→B.λm,x.〈 f(div x m), g(mod x m) 〉. | definition | prodF | matita/matita/lib/arithmetics | matita/matita/lib/arithmetics/bigops.ma | [] | [
"div",
"mod",
"nat"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
bigop (n:nat) (p:nat → bool) (B:Type[0])
(nil: B) (op: B → B → B) (f: nat → B) | ≝
match n with
[ O ⇒ nil
| S k ⇒
match p k with
[true ⇒ op (f k) (bigop k p B nil op f)
|false ⇒ bigop k p B nil op f]
]. | let rec | bigop | matita/matita/lib/arithmetics | matita/matita/lib/arithmetics/bigops.ma | [] | [
"bool",
"nat"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
bigop_Strue: ∀k,p,B,nil,op.∀f:nat→B. p k = true →
\big[op,nil]_{i < S k | p i}(f i) =
op (f k) (\big[op,nil]_{i < k | p i}(f i)). | #k #p #B #nil #op #f #H normalize >H // qed. | lemma | bigop_Strue | matita/matita/lib/arithmetics | matita/matita/lib/arithmetics/bigops.ma | [] | [
"big",
"nat"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
bigop_Sfalse: ∀k,p,B,nil,op.∀f:nat→B. p k = false →
\big[op,nil]_{ i < S k | p i}(f i) =
\big[op,nil]_{i < k | p i}(f i). | #k #p #B #nil #op #f #H normalize >H // qed. | lemma | bigop_Sfalse | matita/matita/lib/arithmetics | matita/matita/lib/arithmetics/bigops.ma | [] | [
"big",
"nat"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
same_bigop : ∀k,p1,p2,B,nil,op.∀f,g:nat→B.
sameF_upto k bool p1 p2 → sameF_p k p1 B f g →
\big[op,nil]_{i < k | p1 i}(f i) =
\big[op,nil]_{i < k | p2 i}(g i). | #k #p1 #p2 #B #nil #op #f #g (elim k) //
#n #Hind #samep #samef normalize >Hind /2/
<(samep … (le_n …)) cases(true_or_false (p1 n)) #H1 >H1
normalize // <(samef … (le_n …) H1) //
qed. | lemma | same_bigop | matita/matita/lib/arithmetics | matita/matita/lib/arithmetics/bigops.ma | [] | [
"big",
"bool",
"nat",
"p1",
"p2",
"sameF_p",
"sameF_upto",
"true_or_false"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
pad_bigop: ∀k,n,p,B,nil,op.∀f:nat→B. n ≤ k →
\big[op,nil]_{i < n | p i}(f i)
= \big[op,nil]_{i < k | if leb n i then false else p i}(f i). | #k #n #p #B #nil #op #f #lenk (elim lenk)
[@same_bigop #i #lti // >(not_le_to_leb_false …) /2/
|#j #leup #Hind >bigop_Sfalse >(le_to_leb_true … leup) //
] qed. | theorem | pad_bigop | matita/matita/lib/arithmetics | matita/matita/lib/arithmetics/bigops.ma | [] | [
"big",
"bigop_Sfalse",
"le_to_leb_true",
"leb",
"nat",
"not_le_to_leb_false",
"same_bigop"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
pad_bigop1: ∀k,n,p,B,nil,op.∀f:nat→B. n ≤ k →
(∀i. n ≤ i → i < k → p i = false) →
\big[op,nil]_{i < n | p i}(f i)
= \big[op,nil]_{i < k | p i}(f i). | #k #n #p #B #nil #op #f #lenk (elim lenk)
[#_ @same_bigop #i #lti //
|#j #leup #Hind #Hfalse >bigop_Sfalse
[@Hind #i #leni #ltij @Hfalse // @le_S //
|@Hfalse //
]
]
qed. | theorem | pad_bigop1 | matita/matita/lib/arithmetics | matita/matita/lib/arithmetics/bigops.ma | [] | [
"big",
"bigop_Sfalse",
"nat",
"same_bigop"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
bigop_false: ∀n,B,nil,op.∀f:nat→B.
\big[op,nil]_{i < n | false }(f i) = nil. | #n #B #nil #op #f elim n // #n1 #Hind
>bigop_Sfalse //
qed. | theorem | bigop_false | matita/matita/lib/arithmetics | matita/matita/lib/arithmetics/bigops.ma | [] | [
"big",
"bigop_Sfalse",
"nat"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
Aop (A:Type[0]) (nil:A) : Type[0] | ≝
{op :2> A → A → A;
nill:∀a. op nil a = a;
nilr:∀a. op a nil = a;
assoc: ∀a,b,c.op a (op b c) = op (op a b) c
}. | record | Aop | matita/matita/lib/arithmetics | matita/matita/lib/arithmetics/bigops.ma | [] | [
"assoc"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
pad_bigop_nil: ∀k,n,p,B,nil.∀op:Aop B nil.∀f:nat→B. n ≤ k →
(∀i. n ≤ i → i < k → p i = false ∨ f i = nil) →
\big[op,nil]_{i < n | p i}(f i)
= \big[op,nil]_{i < k | p i}(f i). | #k #n #p #B #nil #op #f #lenk (elim lenk)
[#_ @same_bigop #i #lti //
|#j #leup #Hind #Hfalse cases (true_or_false (p j)) #Hpj
[>bigop_Strue //
cut (f j = nil)
[cases (Hfalse j leup (le_n … )) // >Hpj #H destruct (H)] #Hfj
>Hfj >nill @Hind #i #leni #ltij
cases (Hfalse i leni (le_S … ltij... | theorem | pad_bigop_nil | matita/matita/lib/arithmetics | matita/matita/lib/arithmetics/bigops.ma | [] | [
"Aop",
"big",
"bigop_Sfalse",
"bigop_Strue",
"nat",
"same_bigop",
"true_or_false"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
bigop_sum: ∀k1,k2,p1,p2,B.∀nil.∀op:Aop B nil.∀f,g:nat→B.
op (\big[op,nil]_{i<k1|p1 i}(f i)) \big[op,nil]_{i<k2|p2 i}(g i) =
\big[op,nil]_{i<k1+k2|if leb k2 i then p1 (i-k2) else p2 i}
(if leb k2 i then f (i-k2) else g i). | #k1 #k2 #p1 #p2 #B #nil #op #f #g (elim k1)
[normalize >nill @same_bigop #i #lti
>(lt_to_leb_false … lti) normalize /2/
|#i #Hind normalize <minus_plus_m_m (cases (p1 i))
>(le_to_leb_true … (le_plus_n …)) normalize <Hind //
<assoc //
]
qed. | theorem | bigop_sum | matita/matita/lib/arithmetics | matita/matita/lib/arithmetics/bigops.ma | [] | [
"Aop",
"assoc",
"big",
"k1",
"le_plus_n",
"le_to_leb_true",
"leb",
"lt_to_leb_false",
"minus_plus_m_m",
"nat",
"p1",
"p2",
"same_bigop"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 | |
plus_minus1: ∀a,b,c. c ≤ b → a + (b -c) = a + b -c. | #a #b #c #lecb @sym_eq @plus_to_minus >(commutative_plus c)
>associative_plus <plus_minus_m_m //
qed. | lemma | plus_minus1 | matita/matita/lib/arithmetics | matita/matita/lib/arithmetics/bigops.ma | [] | [
"associative_plus",
"commutative_plus",
"plus_minus_m_m",
"plus_to_minus",
"sym_eq"
] | https://github.com/LPCIC/matita | 794ed25e6e608b2136ce7fa2963bca4115c7e175 |
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